| L(s) = 1 | − 4.49·2-s − 9.16·3-s + 12.2·4-s + 14.5·5-s + 41.2·6-s − 13.8·7-s − 19.0·8-s + 56.9·9-s − 65.4·10-s − 112.·12-s + 13·13-s + 62.1·14-s − 133.·15-s − 12.2·16-s + 131.·17-s − 256.·18-s + 73.2·19-s + 178.·20-s + 126.·21-s − 15.4·23-s + 174.·24-s + 86.9·25-s − 58.4·26-s − 274.·27-s − 169.·28-s + 71.0·29-s + 599.·30-s + ⋯ |
| L(s) = 1 | − 1.59·2-s − 1.76·3-s + 1.52·4-s + 1.30·5-s + 2.80·6-s − 0.746·7-s − 0.841·8-s + 2.10·9-s − 2.07·10-s − 2.69·12-s + 0.277·13-s + 1.18·14-s − 2.29·15-s − 0.191·16-s + 1.87·17-s − 3.35·18-s + 0.884·19-s + 1.99·20-s + 1.31·21-s − 0.139·23-s + 1.48·24-s + 0.695·25-s − 0.441·26-s − 1.95·27-s − 1.14·28-s + 0.455·29-s + 3.65·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 2 | \( 1 + 4.49T + 8T^{2} \) |
| 3 | \( 1 + 9.16T + 27T^{2} \) |
| 5 | \( 1 - 14.5T + 125T^{2} \) |
| 7 | \( 1 + 13.8T + 343T^{2} \) |
| 17 | \( 1 - 131.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 73.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 15.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 71.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 100.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 84.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 152.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 267.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 402.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 1.78T + 1.48e5T^{2} \) |
| 59 | \( 1 - 872.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 352.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 753.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.12e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.05e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.25e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 145.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 911.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 723.T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.980416596776566108206233831183, −7.78663930596662390948824615651, −6.99150013338939589084636050461, −6.28202028779949020785564902030, −5.71055799254378424000031379209, −4.95954046566901557626592694290, −3.25357782601214815929634634677, −1.68326546059474559818786522017, −1.06042480019168898337018711335, 0,
1.06042480019168898337018711335, 1.68326546059474559818786522017, 3.25357782601214815929634634677, 4.95954046566901557626592694290, 5.71055799254378424000031379209, 6.28202028779949020785564902030, 6.99150013338939589084636050461, 7.78663930596662390948824615651, 8.980416596776566108206233831183
